The Annals of Statistics
- Ann. Statist.
- Volume 34, Number 3 (2006), 1331-1374.
Convergence of algorithms for reconstructing convex bodies and directional measures
We investigate algorithms for reconstructing a convex body K in ℝn from noisy measurements of its support function or its brightness function in k directions u1,…,uk. The key idea of these algorithms is to construct a convex polytope Pk whose support function (or brightness function) best approximates the given measurements in the directions u1,…,uk (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian.
It is shown that this procedure is (strongly) consistent, meaning that, almost surely, Pk tends to K in the Hausdorff metric as k→∞. Here some mild assumptions on the sequence (ui) of directions are needed. Using results from the theory of empirical processes, estimates of rates of convergence are derived, which are first obtained in the L2 metric and then transferred to the Hausdorff metric. Along the way, a new estimate is obtained for the metric entropy of the class of origin-symmetric zonoids contained in the unit ball.
Similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in k directions u1,…,uk. Here the Dudley and Prohorov metrics are used. The methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body.
Ann. Statist., Volume 34, Number 3 (2006), 1331-1374.
First available in Project Euclid: 10 July 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 62M30: Spatial processes 65D15: Algorithms for functional approximation
Secondary: 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G10: Stationary processes
Convex body convex polytope support function brightness function surface area measure least squares set-valued estimator cosine transform algorithm geometric tomography stereology fiber process directional measure rose of intersections
Gardner, Richard J.; Kiderlen, Markus; Milanfar, Peyman. Convergence of algorithms for reconstructing convex bodies and directional measures. Ann. Statist. 34 (2006), no. 3, 1331--1374. doi:10.1214/009053606000000335. https://projecteuclid.org/euclid.aos/1152540751